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proving monad-law-1 ...
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Tue, 25 Nov 2014 17:33:06 +0900
parents 0f308ddd6136
children 15eec529dfc4
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5ba82f107a95 Define Similar in Agda
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1 open import list
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6e6d646d7722 Split basic functions to file
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2 open import basic
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e0ba1bf564dd Apply level to some functions
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3
e0ba1bf564dd Apply level to some functions
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4 open import Level
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742e62fc63e4 Define Monad-law 1-4
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5 open import Relation.Binary.PropositionalEquality
742e62fc63e4 Define Monad-law 1-4
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6 open ≡-Reasoning
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7
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8 module delta where
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9
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11 data Delta {l : Level} (A : Set l) : (Set (suc l)) where
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12 mono : A -> Delta A
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13 delta : A -> Delta A -> Delta A
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14
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15 deltaAppend : {l : Level} {A : Set l} -> Delta A -> Delta A -> Delta A
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16 deltaAppend (mono x) d = delta x d
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17 deltaAppend (delta x d) ds = delta x (deltaAppend d ds)
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18
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19 headDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
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20 headDelta (mono x) = mono x
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21 headDelta (delta x _) = mono x
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23 tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
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24 tailDelta (mono x) = mono x
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25 tailDelta (delta _ d) = d
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5ba82f107a95 Define Similar in Agda
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6ce83b2c9e59 Proof Functor-laws
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27
6ce83b2c9e59 Proof Functor-laws
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28 -- Functor
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29 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Delta A) -> (Delta B)
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30 fmap f (mono x) = mono (f x)
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31 fmap f (delta x d) = delta (f x) (fmap f d)
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35 -- Monad (Category)
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36 eta : {l : Level} {A : Set l} -> A -> Delta A
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37 eta x = mono x
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742e62fc63e4 Define Monad-law 1-4
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38
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39 bind : {l ll : Level} {A : Set l} {B : Set ll} -> (Delta A) -> (A -> Delta B) -> Delta B
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40 bind (mono x) f = f x
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41 bind (delta x d) f = deltaAppend (headDelta (f x)) (bind d (tailDelta ∙ f))
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42
46b15f368905 Define bind and mu for Infinite Delta
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43 mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A
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44 mu d = bind d id
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46b15f368905 Define bind and mu for Infinite Delta
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46
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47 returnS : {l : Level} {A : Set l} -> A -> Delta A
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48 returnS x = mono x
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49
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50 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A
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51 returnSS x y = deltaAppend (returnS x) (returnS y)
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0bc402f970b3 Proof Monad-law 1
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53
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54 -- Monad (Haskell)
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55 return : {l : Level} {A : Set l} -> A -> Delta A
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56 return = eta
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57
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23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
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58 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} ->
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59 (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B)
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60 (mono x) >>= f = f x
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61 (delta x d) >>= f = deltaAppend (headDelta (f x)) (d >>= (tailDelta ∙ f))
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a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
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63
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6ce83b2c9e59 Proof Functor-laws
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64
6ce83b2c9e59 Proof Functor-laws
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65 -- proofs
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67 -- sub proofs
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68
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69 head-delta-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll} ->
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70 (f : A -> B) (d : Delta A) -> (headDelta (fmap f d)) ≡ fmap f (headDelta d)
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71 head-delta-natural-transformation f (mono x) = refl
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72 head-delta-natural-transformation f (delta x d) = refl
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73
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74 tail-delta-natural-transfomation : {l ll : Level} {A : Set l} {B : Set ll} ->
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75 (f : A -> B) (d : Delta A) -> (tailDelta (fmap f d)) ≡ fmap f (tailDelta d)
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76 tail-delta-natural-transfomation f (mono x) = refl
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77 tail-delta-natural-transfomation f (delta x d) = refl
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78
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79 delta-append-natural-transfomation : {l ll : Level} {A : Set l} {B : Set ll} ->
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80 (f : A -> B) (d : Delta A) (dd : Delta A) ->
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81 deltaAppend (fmap f d) (fmap f dd) ≡ fmap f (deltaAppend d dd)
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82 delta-append-natural-transfomation f (mono x) dd = refl
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83 delta-append-natural-transfomation f (delta x d) dd = begin
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84 deltaAppend (fmap f (delta x d)) (fmap f dd)
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85 ≡⟨ refl ⟩
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86 deltaAppend (delta (f x) (fmap f d)) (fmap f dd)
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87 ≡⟨ refl ⟩
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88 delta (f x) (deltaAppend (fmap f d) (fmap f dd))
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89 ≡⟨ cong (\d -> delta (f x) d) (delta-append-natural-transfomation f d dd) ⟩
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90 delta (f x) (fmap f (deltaAppend d dd))
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91 ≡⟨ refl ⟩
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92 fmap f (deltaAppend (delta x d) dd)
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93 ∎
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474ed34e4f02 proving monad-law-1 ...
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94 {-
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95
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96 mu-head-delta : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> mu (headDelta d) ≡ headDelta (mu d)
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97 mu-head-delta (mono (mono x)) = refl
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98 mu-head-delta (mono (delta x (mono xx))) = begin
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99 mu (headDelta (mono (delta x (mono xx))))
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100 ≡⟨ refl ⟩
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101 bind (headDelta (mono (delta x (mono xx)))) id
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102 ≡⟨ refl ⟩
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103 bind (delta x (mono xx)) return
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104 ≡⟨ refl ⟩
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105 deltaAppend (headDelta (return x)) (bind (mono xx) (tailDelta ∙ return))
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106 ≡⟨ refl ⟩
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107 deltaAppend (headDelta (return x)) ((tailDelta ∙ return) xx)
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108 ≡⟨ refl ⟩
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109 deltaAppend (headDelta (mono x)) (tailDelta (mono xx))
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110 ≡⟨ refl ⟩
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111 deltaAppend (mono x) (mono xx)
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112 ≡⟨ refl ⟩
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113 delta x (mono xx)
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114 ≡⟨ {!!} ⟩
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diff changeset
115 headDelta (delta x (mono xx))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
116 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
117 headDelta (bind (mono (delta x (mono xx))) id)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
118 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
119 headDelta (mu (mono (delta x (mono xx))))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
120 ∎
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
121 mu-head-delta (mono (delta x (delta x₁ d))) = {!!}
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
122 mu-head-delta (delta d dd) = {!!}
63
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
123 -}
38
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
124 -- Functor-laws
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
125
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
126 -- Functor-law-1 : T(id) = id'
55
9c8c09334e32 Redefine Delta for infinite changes in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 54
diff changeset
127 functor-law-1 : {l : Level} {A : Set l} -> (d : Delta A) -> (fmap id) d ≡ id d
57
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
128 functor-law-1 (mono x) = refl
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
129 functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d)
38
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
130
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
131 -- Functor-law-2 : T(f . g) = T(f) . T(g)
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
132 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
55
9c8c09334e32 Redefine Delta for infinite changes in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 54
diff changeset
133 (f : B -> C) -> (g : A -> B) -> (d : Delta A) ->
9c8c09334e32 Redefine Delta for infinite changes in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 54
diff changeset
134 (fmap (f ∙ g)) d ≡ ((fmap f) ∙ (fmap g)) d
57
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
135 functor-law-2 f g (mono x) = refl
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
136 functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d)
38
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
137
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
138 -- Monad-laws (Category)
63
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
139
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
140 monad-law-1-4 : {l : Level} {A : Set l} -> (ds : Delta (Delta A)) ->
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
141 tailDelta (bind ds (tailDelta ∙ id)) ≡ bind (tailDelta ds) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
142 monad-law-1-4 (mono ds) = refl
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
143 monad-law-1-4 (delta (mono x) ds₁) = refl
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
144 monad-law-1-4 (delta (delta x (mono x₁)) ds₁) = refl
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
145 monad-law-1-4 (delta (delta x (delta x₁ ds)) ds₁) = refl
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
146
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
147 monad-law-1-3 : {l : Level} {A : Set l} -> (ds : Delta (Delta A)) ->
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
148 tailDelta (bind ds tailDelta) ≡ bind (tailDelta ds) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
149 monad-law-1-3 (mono ds) = refl
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
150 monad-law-1-3 (delta (mono x) ds) = refl
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
151 monad-law-1-3 (delta (delta x (mono x₁)) ds) = refl
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
152 monad-law-1-3 (delta (delta x (delta x₁ d)) ds) = refl
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
153
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
154 monad-law-1-sub-sub : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) ->
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
155 bind (fmap mu d) (tailDelta ∙ tailDelta) ≡ bind (bind d (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
156 monad-law-1-sub-sub (mono (mono d)) = refl
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
157 monad-law-1-sub-sub (mono (delta (mono x) ds)) = begin
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
158 bind (fmap mu (mono (delta (mono x) ds))) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
159 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
160 bind (mono (mu (delta (mono x) ds))) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
161 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
162 bind (mono (bind (delta (mono x) ds) id)) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
163 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
164 bind (mono (deltaAppend (headDelta (mono x)) (bind ds tailDelta))) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
165 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
166 bind (mono (deltaAppend (mono x) (bind ds tailDelta))) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
167 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
168 bind (mono (delta x (bind ds tailDelta))) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
169 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
170 (tailDelta ∙ tailDelta) (delta x (bind ds tailDelta))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
171 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
172 tailDelta (bind ds tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
173 ≡⟨ monad-law-1-3 ds ⟩ -- ?
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
174 bind (tailDelta ds) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
175 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
176 bind ((tailDelta ∙ tailDelta) (delta (mono x) ds)) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
177 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
178 bind (bind (mono (delta (mono x) ds)) (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
179 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
180 bind (bind (headDelta (tailDelta (mono (delta (mono x) ds)))) (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
181 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
182 bind (bind (mono (delta (mono x) ds)) (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
183 ∎
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
184 monad-law-1-sub-sub (mono (delta (delta x (mono x₁)) ds)) = begin
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
185 bind (fmap mu (mono (delta (delta x (mono x₁)) ds))) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
186 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
187 bind (mono (mu (delta (delta x (mono x₁)) ds))) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
188 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
189 (tailDelta ∙ tailDelta) (mu (delta (delta x (mono x₁)) ds))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
190 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
191 (tailDelta ∙ tailDelta) (bind (delta (delta x (mono x₁)) ds) id)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
192 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
193 (tailDelta ∙ tailDelta) (deltaAppend (headDelta (delta x (mono x₁))) (bind ds (tailDelta ∙ id)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
194 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
195 (tailDelta ∙ tailDelta) (deltaAppend (mono x) (bind ds (tailDelta ∙ id)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
196 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
197 (tailDelta ∙ tailDelta) (delta x (bind ds (tailDelta ∙ id)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
198 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
199 tailDelta (bind ds (tailDelta ∙ id))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
200 ≡⟨ monad-law-1-4 ds ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
201 bind (tailDelta ds) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
202 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
203 bind ((tailDelta ∙ tailDelta) (delta (delta x (mono x₁)) ds)) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
204 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
205 bind (bind (mono (delta (delta x (mono x₁)) ds)) (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
206 ∎
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
207 monad-law-1-sub-sub (mono (delta (delta x (delta xx d)) ds)) = begin
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
208 bind (fmap mu (mono (delta (delta x (delta xx d)) ds))) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
209 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
210 bind (mono (mu (delta (delta x (delta xx d)) ds))) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
211 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
212 (tailDelta ∙ tailDelta) (mu (delta (delta x (delta xx d)) ds))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
213 ≡⟨ {!!} ⟩ -- ?
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
214 bind (bind (mono (delta (delta x (delta xx d)) ds)) (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
215 ∎
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
216 monad-law-1-sub-sub (delta d ds) = {!!}
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
217
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
218
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
219 monad-law-1-sub : {l : Level } {A : Set l} -> (x : Delta (Delta A)) -> (d : Delta (Delta (Delta A))) ->
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
220 deltaAppend (headDelta (mu x)) (bind (fmap mu d) tailDelta) ≡ mu (deltaAppend (headDelta x) (bind d tailDelta))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
221 monad-law-1-sub (mono (mono _)) (mono (mono _)) = refl
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
222 monad-law-1-sub (mono (mono _)) (mono (delta (mono _) _)) = refl
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
223 monad-law-1-sub (mono (mono _)) (mono (delta (delta _ _) _)) = refl
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
224 monad-law-1-sub (mono (mono x)) (delta (mono (mono xx)) d) = begin
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
225 deltaAppend (headDelta (mu (mono (mono x)))) (bind (fmap mu (delta (mono (mono xx)) d)) tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
226 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
227 deltaAppend (headDelta (mu (mono (mono x)))) (bind (delta (mu (mono (mono xx))) (fmap mu d)) tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
228 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
229 deltaAppend (headDelta (bind (mono (mono x)) id)) (bind (delta (mu (mono (mono xx))) (fmap mu d)) tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
230 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
231 deltaAppend (headDelta (mono x)) (bind (delta (mu (mono (mono xx))) (fmap mu d)) tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
232 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
233 deltaAppend (headDelta (mono x)) (bind (delta (mono xx) (fmap mu d)) tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
234 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
235 deltaAppend (mono x) (bind (delta (mono xx) (fmap mu d)) tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
236 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
237 deltaAppend (mono x) (bind (delta (mono xx) (fmap mu d)) tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
238 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
239 deltaAppend (mono x) (deltaAppend (tailDelta (mono xx)) (bind (fmap mu d) (tailDelta ∙ tailDelta)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
240 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
241 deltaAppend (mono x) (deltaAppend (mono xx) (bind (fmap mu d) (tailDelta ∙ tailDelta)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
242 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
243 deltaAppend (mono x) (deltaAppend (mu (mono (mono xx))) (bind (fmap mu d) (tailDelta ∙ tailDelta)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
244 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
245 deltaAppend (mono x) (deltaAppend (mono xx) (bind (fmap mu d) (tailDelta ∙ tailDelta)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
246 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
247 delta x (deltaAppend (mono xx) (bind (fmap mu d) (tailDelta ∙ tailDelta)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
248 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
249 delta x (delta xx (bind (fmap mu d) (tailDelta ∙ tailDelta)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
250 ≡⟨ cong (\d -> (delta x (delta xx d))) (monad-law-1-sub-sub d) ⟩ -- ???
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
251 delta x (delta xx (bind (bind d (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
252 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
253 delta x ((deltaAppend (mono xx) (bind (bind d (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta))))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
254 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
255 delta x ((deltaAppend (tailDelta (mono xx)) (bind (bind d (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta))))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
256 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
257 delta x (bind (delta (mono xx) (bind d (tailDelta ∙ tailDelta))) tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
258 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
259 delta x (bind (deltaAppend (mono (mono xx)) (bind d (tailDelta ∙ tailDelta))) tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
260 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
261 delta x (bind (deltaAppend (headDelta (tailDelta (mono (mono xx)))) (bind d (tailDelta ∙ tailDelta))) tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
262 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
263 delta x (bind (bind (delta (mono (mono xx)) d) tailDelta) tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
264 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
265 deltaAppend (mono x) (bind (bind (delta (mono (mono xx)) d) tailDelta) tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
266 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
267 bind (delta (mono x) (bind (delta (mono (mono xx)) d) tailDelta)) id
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
268 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
269 mu (delta (mono x) (bind (delta (mono (mono xx)) d) tailDelta))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
270 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
271 mu (deltaAppend (mono (mono x)) (bind (delta (mono (mono xx)) d) tailDelta))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
272 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
273 mu (deltaAppend (headDelta (mono (mono x))) (bind (delta (mono (mono xx)) d) tailDelta))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
274 ∎
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
275 monad-law-1-sub (mono (mono x)) (delta (mono (delta x₁ d)) d₁) = {!!}
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
276 monad-law-1-sub (mono (mono x)) (delta (delta d d₁) d₂) = {!!}
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
277 monad-law-1-sub (mono (delta x x₁)) d = {!!}
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
278 monad-law-1-sub (delta x x₁) d = {!!}
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
279
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
280 -- monad-law-1 : join . fmap join = join . join
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
281 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
282 monad-law-1 (mono d) = refl
63
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
283 monad-law-1 (delta x d) = begin
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
284 (mu ∙ (fmap mu)) (delta x d)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
285 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
286 mu (fmap mu (delta x d))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
287 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
288 mu (delta (mu x) (fmap mu d))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
289 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
290 bind (delta (mu x) (fmap mu d)) id
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
291 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
292 deltaAppend (headDelta (mu x)) (bind (fmap mu d) tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
293 ≡⟨ monad-law-1-sub x d ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
294 mu (deltaAppend (headDelta x) (bind d tailDelta))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
295 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
296 mu (bind (delta x d) id)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
297 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
298 mu (mu (delta x d))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
299 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
300 (mu ∙ mu) (delta x d)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
301 ∎
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
302
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
303 -- split d
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
304 {-
62
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
305 monad-law-1 (delta x (mono d)) = begin
63
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
306
62
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
307 (mu ∙ fmap mu) (delta x (mono d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
308 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
309 mu (fmap mu (delta x (mono d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
310 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
311 mu (delta (mu x) (mono (mu d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
312 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
313 bind (delta (mu x) (mono (mu d))) id
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
314 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
315 deltaAppend (headDelta (mu x)) (bind (mono (mu d)) tailDelta)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
316 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
317 deltaAppend (headDelta (mu x)) (tailDelta (mu d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
318 ≡⟨ {!!} ⟩
63
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
319 mu (deltaAppend (headDelta x) (tailDelta d))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
320 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
321 mu (deltaAppend (headDelta x) (tailDelta (id d)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
322 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
323 mu (deltaAppend (headDelta x) ((tailDelta ∙ id) d))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
324 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
325 mu (deltaAppend (headDelta x) (bind (mono d) (tailDelta ∙ id)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
326 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
327 mu (bind (delta x (mono d)) id)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
328 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
329 mu (mu (delta x (mono d)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
330 ≡⟨ refl ⟩
62
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
331 (mu ∙ mu) (delta x (mono d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
332 ∎
63
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
333 monad-law-1 (delta x (delta d ds)) = begin
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
334 (mu ∙ fmap mu) (delta x (delta d ds))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
335 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
336 mu (fmap mu (delta x (delta d ds)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
337 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
338 mu (delta (mu x) (delta (mu d) (fmap mu ds)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
339 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
340 bind (delta (mu x) (delta (mu d) (fmap mu ds))) id
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
341 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
342 deltaAppend (headDelta (mu x)) (bind (delta (mu d) (fmap mu ds)) tailDelta)
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
343 ≡⟨ refl ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
344 deltaAppend (headDelta (mu x)) (deltaAppend (headDelta (tailDelta (mu d))) (bind (fmap mu ds) (tailDelta ∙ tailDelta)))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
345
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
346 ≡⟨ {!!} ⟩
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
347 (mu ∙ mu) (delta x (delta d ds))
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
348 ∎
62
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
349 -}
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
350
29
e0ba1bf564dd Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
351
62
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
352 {-
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
353 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
354 monad-law-1 (mono d) = refl
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
355 monad-law-1 (delta x (mono d)) = begin
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
356 (mu ∙ fmap mu) (delta x (mono d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
357 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
358 mu ((fmap mu) (delta x (mono d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
359 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
360 mu (delta (mu x) (fmap mu (mono d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
361 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
362 mu (delta (mu x) (fmap mu (mono d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
363 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
364 mu (delta (mu x) (mono (mu d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
365 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
366 bind (delta (mu x) (mono (mu d))) id
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
367 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
368 deltaAppend (headDelta (mu x)) (bind (mono (mu d)) (tailDelta ∙ id))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
369 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
370 deltaAppend (headDelta (mu x)) (bind (mono (mu d)) (tailDelta))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
371 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
372 deltaAppend (headDelta (mu x)) (tailDelta (mu d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
373 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
374 deltaAppend (headDelta (mu x)) ((tailDelta ∙ mu) d)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
375 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
376 deltaAppend (headDelta (mu x)) (bind (mono d) (tailDelta ∙ mu))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
377 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
378 bind (delta x (mono d)) mu
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
379 ≡⟨ {!!} ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
380 mu (deltaAppend (headDelta x) (tailDelta d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
381 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
382 mu (deltaAppend (headDelta x) (bind (mono d) tailDelta))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
383 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
384 mu (deltaAppend (headDelta (id x)) (bind (mono d) (tailDelta ∙ id)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
385 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
386 mu (deltaAppend (headDelta x) (bind (mono d) (tailDelta ∙ id)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
387 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
388 mu (bind (delta x (mono d)) id)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
389 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
390 mu (deltaAppend (headDelta (id x)) (bind (mono d) (tailDelta ∙ id)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
391 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
392 mu (mu (delta x (mono d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
393 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
394 (mu ∙ mu) (delta x (mono d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
395 ∎
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
396 monad-law-1 (delta x (delta xx d)) = {!!}
63
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
397
62
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
398 monad-law-1 (delta x d) = begin
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
399 (mu ∙ fmap mu) (delta x d)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
400 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
401 mu ((fmap mu) (delta x d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
402 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
403 mu (delta (mu x) (fmap mu d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
404 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
405 bind (delta (mu x) (fmap mu d)) id
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
406 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
407 deltaAppend (headDelta (mu x)) (bind (fmap mu d) (tailDelta ∙ id))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
408 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
409 deltaAppend (headDelta (mu x)) (bind (fmap mu d) (tailDelta ∙ id))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
410 ≡⟨ {!!} ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
411 (mu ∙ mu) (delta x d)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
412 ∎
63
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
413
34
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
414
63
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
415
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
416
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
417 -- monad-law-2-2 : join . return = id
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
418 monad-law-2-2 : {l : Level} {A : Set l } -> (s : Delta A) -> (mu ∙ eta) s ≡ id s
35
c5cdbedc68ad Proof Monad-law-2-2
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
419 monad-law-2-2 (similar lx x ly y) = refl
c5cdbedc68ad Proof Monad-law-2-2
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
420
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
421 -- monad-law-3 : return . f = fmap f . return
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
422 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x
36
169ec60fcd36 Proof Monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 35
diff changeset
423 monad-law-3 f x = refl
27
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
424
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
425 -- monad-law-4 : join . fmap (fmap f) = fmap f . join
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
426 monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (s : Delta (Delta A)) ->
36
169ec60fcd36 Proof Monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 35
diff changeset
427 (mu ∙ fmap (fmap f)) s ≡ (fmap f ∙ mu) s
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
428 monad-law-4 f (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = refl
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
429
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
430
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
431 -- Monad-laws (Haskell)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
432 -- monad-law-h-1 : return a >>= k = k a
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
433 monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} ->
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
434 (a : A) -> (k : A -> (Delta B)) ->
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
435 (return a >>= k) ≡ (k a)
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
436 monad-law-h-1 a k = refl
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
437
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
438
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
439
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
440 -- monad-law-h-2 : m >>= return = m
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
441 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
442 monad-law-h-2 (mono x) = refl
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
443 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d)
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
444
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
445
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
446
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
447
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
448 -- monad-law-h-3 : m >>= (\x -> k x >>= h) = (m >>= k) >>= h
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
449 monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
450 (m : Delta A) -> (k : A -> (Delta B)) -> (h : B -> (Delta C)) ->
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
451 (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h)
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
452 monad-law-h-3 (mono x) k h = refl
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
453 monad-law-h-3 (delta x d) k h = begin
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
454 (delta x d) >>= (\x -> k x >>= h)
42
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
455 ≡⟨ refl ⟩
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
456 -- (delta x d) >>= f = deltaAppend (headDelta (f x)) (d >>= (tailDelta ∙ f))
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
457 deltaAppend (headDelta ((\x -> k x >>= h) x)) (d >>= (tailDelta ∙ (\x -> k x >>= h)))
42
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
458 ≡⟨ refl ⟩
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
459 deltaAppend (headDelta (k x >>= h)) (d >>= (tailDelta ∙ (\x -> k x >>= h)))
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
460 ≡⟨ {!!} ⟩
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
461 ((delta x d) >>= k) >>= h
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
462 ∎
63
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
463 -}